You may be on the incorrect tax code at the moment, since you were paid before you got your NI number.
What is the tax code on your payslip?
If it’s BR, you’ll be on Basic Rate. If it ends in W1, M1 or X, you’ll be on an emergency code.
The tax code most people will have is 1150L, though I think it ends with M or N for marriage allowances.
If you’re on BR or an emergency tax code that’s probably why you have NI category A instead of B.
If this is the case, I would give your tax office a call and ask them to correct your tax code.
I’m British but I ended up on the BR tax code for about a year once because they didn’t have my details correct and I didn’t even notice. I called them and they sorted it out really quickly.
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